A family of neighbourly polytopes in R2d with N=2d+4 vertices is constructed. All polytopes in the family have a planar Gale diagram of a special type, namely, with exactly d+3 black points in convex position. These Gale diagrams are parametrized by 3-trees (trees with a certain additional structure). For all polytopes in the family, the number of faces of dimension m containing a given vertex A depends only on d and m.
For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X.
We put forward a method for constructing semiorthogonal decompositions of the derived category of G-equivariant sheaves on a variety X under the assumption that the derived category of sheaves on X admits a semiorthogonal decomposition with components preserved by the action of the group G on X. This method is used to obtain semiorthogonal decompositions of equivariant derived categories for projective bundles and blow-ups with a smooth centre as well as for varieties with a full exceptional collection preserved by the group action. Our main technical tool is descent theory for derived categories.
The aim of this paper is to prove ergodic decomposition theorems for probability measures which are quasi-invariant under Borel actions of inductively compact groups as well as for σ-finite invariant measures. For infinite measures the ergodic decomposition is not unique, but the measure class of the decomposing measure on the space of projective measures is uniquely defined by the initial invariant measure.